important probability distributions

TUM School of Management Production and Supply Chain Management Prof Martin Grunow Technische Universitt Mnchen

Quality Engineering & Management

Session 2.3: Important Probability Distributions

Dr. Holly Ott Production and Supply Chain Management

Chair: Prof. Martin Grunow TUM School of Management

Holly Ott Quality Engineering & Management Module 2.3 1


Learning Objectives

Discuss the nature Binomial and Poisson probability distributions for discrete random variables, the context in which they are useful, and their important characteristics.

Calculate the probability of a given event using Binomial and Poisson probability distributions.

Describe the Normal distribution for continuous random variables.



Some Important Probability Distributions

Binomial Distribution discrete Poisson Distribution discrete Normal Distribution continuous


2012 from "A First Course in Quality Engineering: Integrating Statistical and Management Methods of Quality" by K.S. Krishnamoorthi. Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.


Combinations Before we start, we need to know the number of combinations of n

distinct objects taken r at a time written as is given by: n! = "n factorial" = 1 for n = 0

= 1 2 3 n for n 1




The Binomial Distribution


A random variable X is said to have the binomial distribution with parameters n and p if its probability distribution is given by: We write X ~ Bi(n,p) to indicate X has a binomial distribution. X represents the number of successes of n independent trials, where p is the probability of success and (1 - p) is the probability of failure in one trial. Parameter p: 0 < p < 1


( ) ,..., n, , xppxn

p(x)= xnx 101 =




Reiner Hutwelker

( ) ,..., n, , xppxn

p(x)= xnx 101 =




( ) ,..., n, , xppxn

p(x)= xnx 101 =

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

(n=80, p=0.1)

(n=80, p=0.2)

(n=30, p=0.1)

(n=30, p=0.2)

Argon Chen


Examples of Binomial R.Vs.


1. X: The number of heads when a fair coin is tossed 10 times X ~ Bi(10,1/2)

2. Y: The number of baskets a ball player makes in 12 free throws, if

her average is 0.4 Y ~ Bi(12,0.4)

3. W: The number of defectives in a sample of 20 taken from a large product batch ("lot") having 2% defectives

W ~ Bi(20,0.02)



Calculations with Binomial Distribution

Example: A sample of 12 bolts is picked from a production line and inspected. If the process produces 2% defectives, what is the probability the sample will have exactly 1 defective?

Let X be the number of defectives out of 12. Then: X ~ Bi(12, 0.02)




The Mean and Variance of a Binomial Variable


If X~Bi(n, p), then, using the definition for mean and variance that: and represent the long-run average and standard deviation respectively of the binomial random variable.

=np-p)(pxn

x n-xn

x

xx 1

0=

=

( )p=np-p)(pxn

)(x n-xxn

xxx

=

=

110

22



The Poisson Distribution


A random variable X is said to have the Poisson distribution if its probability mass function is given by:

, x = 0, 1, 2, We write X ~ Po() to indicate X has a Poisson distribution.

( )x!expx

=


Foto: Thommy Weiss / pixelio.de



How to recognize a Poisson variable? Variable is countable and can take values from zero to infinity.

Examples of Poisson random variables:

1. Number of knots per sheet of plywood 2. Number of blemishes per shirt 3. Number of pinholes per square foot of galvanized sheet 4. Number of accidents per month in a factory

We write X ~ Po() to indicate X has a Poisson distribution.






Argon Chen 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

=10

=1

=4

( )x!expx

=

, x = 0, 1, 2,


Poisson Distribution (contd.)

If X = Po(), then the mean and variance of a Poisson variable: Note that, for the Poisson distribution, the mean and variance are equal to the value of the parameter of the distribution.


x = xexx!x=0

=

x2 = (x x )2

x= 0

ex

x! =



Calculating Poisson Probabilities

Example: A typist makes on the average 3 mistakes per page. What is the probability that the page he types for a typing test will have no more than one mistake?

Let X be the number of mistakes per page. X ~ Po(3)




Continuous Distribution Models


Uniform Distribution

Exponential Distribution


The Normal Distribution


A random variable X is said to have the normal distribution with parameters and 2, if its probability density function is given by:

X ~ N(,2)

f x( ) = 1 2 e

12

x

2

, < x < , > 0



The Normal Distribution


1. It is asymptotic with respect to the x-axis 2. It is symmetric with respect to a vertical line at x = 3. The maximum value of f(x) occurs at x = 4. The two points of inflexion occur at distances on each side of

The graph of the normal pdf


Parameters of the Normal Distribution


It can be shown:

[Area under the curve = 1]

[Mean of the distribution = ]

[Variance of the distribution = 2 ] and 2 are the two parameters, mean and variance, of the normal distribution.

( ) =x

dxxf 1

( ) =x

dxxx f

( ) ( ) 22

dxx fxx

=


Coming Up

Lecture 3.1: The Normal Distribution


important probability distributions

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